# Video: Use a P-Test to Find Values for Which a Series Converges

For what values of π does β_(π = 1)^(β) 1/π^(5π) converge?

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### Video Transcript

For what values of π does the sum from π equals one to β of one over π to the five π power converge?

We can, firstly, recognize this series to be a π-series. Recall that a π-series is a series of the form the sum from π equals one to β of one over π to the π power. But to avoid confusion with the π in the question and the π here, letβs replace this π with π. And we have a really useful theorem for π-series, which says that this π-series converges if π is greater than one and diverges otherwise.

So, itβs the value of the exponent π in the denominator, which tells us if the series is convergent. So, for this series to be convergent, five π would need to be greater than one. And we can actually divide both sides through by five to get π greater than one-fifth.

So, what weβre saying is that the sum from π equals one to β of one over π raised to the five π power converges only if π is greater than one-fifth.